Problem 76 - Prealgebra and Introductory Algebra

Question

Divide. $$\frac{8+2 x-x^{2}}{x^{2}+7 x+10} \div \frac{x^{2}-11 x+28}{x^{2}-x-42}$$

Step-by-Step Solution

Answer

The solution to the given algebraic division is \(-\frac{(x-4)(x-5)}{(x+5)(x+7)}\)

Step 1: Rewrite the division as multiplication

To divide rational expressions, we change the division into multiplication by flipping the second fraction. This is known as inverting the second fraction or taking the reciprocal. Our expression will change to: \(\frac{8+2 x-x^{2}}{x^{2}+7 x+10} \times \frac{x^{2}-x-42}{x^{2}-11 x+28}\)

Step 2: Factorization

Factorize all the four numerators and denominators, which makes the rational expressions simpler to multiply by allowing us to cancel out terms that appear in both the numerators and denominators: \[\frac{-(x-2)(x-4)}{(x+2)(x+5)} \times \frac{(x+2)(x-5)}{(x-2)(x+7)}\]

Step 3: Cancel out common factors

After factoring the numerators and denominators, cancel the common factors that appear on both the numerator and denominator: \[\frac{-(x-2)(x-4)}{(x+2)(x+5)} \times \frac{(x+2)(x-5)}{(x-2)(x+7)} = -\frac{(x-4)}{(x+5)} \times \frac{(x-5)}{(x+7)}\]

Step 4: Multiplication

Multiply remaining numerators with numerators and denominators with denominators: \[-\frac{(x-4)(x-5)}{(x+5)(x+7)}\]

Key Concepts

Division of Rational Expressions

When we divide rational expressions, it involves an operation similar to dividing fractions but with algebraic expressions instead of numbers. The core idea is to rewrite the division as a multiplication by flipping the second fraction upside down, which is equivalent to finding the reciprocal.
For example, given an expression like \( \frac{a}{b} \div \frac{c}{d} \), it gets transformed into \( \frac{a}{b} \times \frac{d}{c} \).
This transformation allows the division to proceed as a straightforward multiplication problem.

• Ensure the problem is presented as a division of two fractions.
• Change the division symbol to multiplication.
• Flip the numerator and denominator of the second fraction, making it a reciprocal.

After rewriting the expression, you follow the rules of multiplication for fractions. It’s a crucial technique for simplifying complex algebraic expressions and makes it easier to handle rational expressions.

Multiplication of Rational Expressions

Once you have transformed the division into a multiplication problem, the next step is to multiply the rational expressions. This involves multiplying the numerators together and the denominators together.
In simpler terms, if you have \( \frac{a}{b} \times \frac{c}{d} \), you compute it as \( \frac{a \cdot c}{b \cdot d} \). This operation is straightforward, but it can be even more manageable with some preliminary steps like factorization.

• Multiply the numerators across to form a single numerator.
• Multiply the denominators across to form a single denominator.
• Simplify the resulting expression further if possible.

Always simplify the result by canceling any common factors if they exist. This is especially facilitated by preparing the rational expressions through factorization, ultimately leading to a more concise result.

Factorization in Algebra

Factorization is a crucial step in simplifying rational expressions and is a potent tool in algebra. It involves breaking down expressions into products of simpler factors, revealing opportunities for simplification by canceling terms.
During factorization, you look for common patterns or identities such as difference of squares, perfect square trinomials, or other identifiable linear or quadratic factors.
For instance, factorizing a quadratic expression like \( x^2 - 5x + 6 \), you would find \( (x-2)(x-3) \) as its factors. This is useful for simplifying expressions before multiplication because it allows for canceling of similar terms across numerators and denominators.

• Look for common factors in terms.
• Identify standard factorization patterns (such as \( a^2 - b^2 = (a-b)(a+b) \)).
• Rewrite the expression using their factors for simplification.

This process helps in reducing the complexity of a multiplication problem and makes the entire process more efficient, often leading to cleaner, simplified results which are much easier to interpret.